Learn Common Ratio

In mathematics, a geometric series is a series with a permanent ratio between successive terms. For example, the series 1/2+1/4+1/8+1/16.....is geometric, so each term except the first can be obtained by multiplying the previous term by 1/2.Geometric series are one of the easiest examples of infinite series with finite sums. Basically, geometric series played an important role in the early development of calculus, and they continue to be central in the study of convergence of series. Geometric series(GS) are used throughout mathematics, and they have important applications in physics, engineering, biology, economics, computer science, queueing theory, and finance.

Common ratio:

The terms of a geometric series form a geometric progression(GP), meaning that the ratio of successive terms in the series is constant. Below table shows several geometric series with different common ratios:

Common ratio


Example

10


4 + 40 + 400 + 4000 + 40,000 + ···

1/3


9 + 3 + 1 + 1/3 + 1/9 + ···

1/10


7 + 0.7 + 0.07 + 0.007 + 0.0007 + ···

1


3 + 3 + 3 + 3 + 3 + ···

−1/2


1 − 1/2 + 1/4 − 1/8 + 1/16 − 1/32 + ···

–1


3 − 3 + 3 − 3 + 3 − ···

The behavior of the terms depends on the common ratio r:

If r is between -1 and +1 the terms of the series become smaller and smaller, approaching zero in the limit. The series converges to a sum, as in the case, where r is a half, and the series has the sum one.

If r is greater than 1 or less than -1 the terms of the series become larger and larger. The total sum of the terms also gets larger and larger, and the series has no sum. (The series diverges.)

If r is equal to 1, all of the terms of the series are the same. The series diverges.
If r is minus one the terms take two values alternately (e.g. 2, −2, 2, −2, 2,... ). The sum of the terms oscillates between two values (e.g. 2, 0, 2, 0, 2,... ). This is a different types of divergence and again the series has no sum.

Examples:

Consider the sum of the following geometric series:

s = 1+2/3+4/9+8/27+......

This series has common ratio 2/3. If we multiply through by this common ratio, then the initial one becomes a 2/3, the 2/3 becomes a 4/9, and so on:

2/3 s = 2/3 + 4/9 + 8/27 + 16/81 + .....

Mean Value Theorem

The Mean Value Theorem has very important consequences in differential calculus.

THEOREM: Let the function f such that

i.           continuous in the closed interval [a,b]
ii.            derivable in open interval (a,b)

Then there exists at least one c with  a < c < b such that

calculus formula

The result in the theorem can be expressed as a statement about graph of f: if A(a , f(a)) and B(b , f(b)), are the end points on the graph, then there is at least one point C between A and B,such that the tangent is drawn from C is parallel to the chord AB.

Mean value theorem graph

Mean value theorem is also known as Lagrange’s Mean Value Theorem or First Mean Value Theorem or Law of Mean.

Applications of mean value theorem

1. Let the function be f such that

(i)                  Continuous in interval [a,b]

(ii)                Derivable in interval (a,b)

(iii)              f'(x) = 0 `AA` x  `epsi` (a,b) , then f(x) is constant in [a , b].

2.Let f and g be a functions such that

(i)             f and g are continuous in interval [a,b]

(ii)            f and g are derivable in interval(a,b)

(iii)          f'(x) = g'(x) `AA` x  `epsi`  (a,b) , then f(x) - g(x) is constant in [a,b]

3.Let the function be f such that

(i)             Continuous in interval[a,b]

(ii)            Derivable in interval(a,b)

(iii)          f'(x) > 0 `AA` x  `epsi`  (a,b), then f(x) is strictly increasing function in [a,b]

4.Let the function be f such that

(i)             Continuous in interval[a,b]

(ii)            Derivable in interval (a,b)

(iii)          f'(x) < 0 `AA` x  `epsi`    (a,b), thenf(x)  is strictly decreasing function in[a,b]

Special case of Mean Value Theorem is when f(a) = f(b).Then there exists at least one c with  a < c < b such that f'(c)= 0 . This case is known as Rolle’s Theorem.

Cauchy’s mean value theorem in calculus

Let f and g be functions such that

i.            both are continuous in closed interval [a,b]
ii.            both are derivable in open interval (a,b)
iii.             g'(x) `!=` 0 for any x `epsi` (a,b)  then there exists at least one number c `epsi` (a,b) such that

`(f'(c))/(g'(c))`  =  `(f(b) - f(a))/(g(b) - g(a))`

Mean value theorem example



Verify Rolle's theorem for the function

f (x) = x2 - 8x + 12 on (2, 6)

Since a polynomial function is continuous and differentiable everywhere f (x) is differentiable and continuous (i) and (ii) conditions of Rolle's theorem is satisfied.

f (2) = 22 - 8 (2) + 12 = 0

f (6) = 36 - 48 + 12 = 0

Therefore (iii) condition is satisfied.

Rolle's theorem is applicable for the given function f (x).

\ There must exist c  (2, 6) such that f '(c) = 0

f '(x) = 2x - 8

`=>`  c = 4 `in`(2,6)

Rolle's theorem is verified.